Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Deformation of Honeycombs and Foams Derivation of Young modulus in highly porous foam

Derivation of Young modulus in highly porous foam

Here we estimate the Young modulus of an open-cell foam, represented schematically as shown below, where each cubic cell has a side length l and is made of struts with a square cross-section of thickness t, whose Young modulus is ES. The derivation is similar to that used for the honeycomb. However because the cell is cubic rather than hexagonal, θ = 0, giving the deflection of a single half-beam, δ, as

\[\delta = \frac{1}{{24}}\frac{{F{l^3}}}{{{E_{\rm{S}}}I}}\]

where I is the second moment of area, where I = t4/12. This gives the deflection in the loading direction, Δx, as

\[\Delta x = 2\delta = \frac{F}{{{E_{\rm{S}}}t}}{\rm{ }}{\left( {\frac{l}{t}} \right)^3}\]

This gives the strain, ε, as

\[\varepsilon = \frac{F}{{{E_{\rm{S}}}}}{\rm{ }} \cdot \frac{{{l^2}}}{{{t^4}}}\]

The stress, σ, arising from the applied force F is

\[\sigma = \frac{F}{{{l^2}}}\]

This gives an expression for the Young modulus of the open-cell foam, E

\[E = {E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^4}\]


\[\frac{\rho }{{{\rho _{\rm{S}}}}} \propto {\left( {\frac{t}{l}} \right)^2}\]

The relative elastic modulus of the open-cell foam, E/ES, is therefore related to the relative density, ρ/ρS, according to

\[\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}\]

where k is a constant approximately equal to 1.